Restricted maximum likelihood estimation for animal models using derivatives of the likelihood

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Restricted maximum likelihood estimation for animal

models using derivatives of the likelihood

K Meyer, Sp Smith

To cite this version:

Original

article

Restricted maximum likelihood

estimation for animal models

_using

derivatives

of

the likelihood

K

_Meyer,

SP Smith

Animal Genetics and

_{Breeding Unit, University of}

New

_{England, Armidale,}

NSW

_2351,

Australia

(Received

21 March

_{1995; accepted}

9 October

₁₉₉₅₎

Summary - Restricted maximum likelihood estimation

_using

first and second derivatives of the likelihood is described. It relies on the calculation of derivatives without the need

for

_large

matrix inversion

_using

an automatic differentiation

_procedure.

In _essence, this

is an extension of the

Cholesky

factorisation of a matrix. A

_{reparameterisation}

is used

to transform the constrained

_optimisation

_{problem imposed}

in

_estimating

covariance components to an unconstrained

problem,

thus

making

the use of

Newton-Raphson

and related

algorithms

feasible. A numerical

example

is

_given

to illustrate calculations. Several modified

_{Newton-Raphson}

and method of

_{scoring algorithms}

are

_compared

for

applications

to

_analyses

of beef cattle

_data,

and contrasted to a derivative-free

_algorithm.

restricted maximum likelihood

_/

derivative

_/

_algorithm

_/

variance _component esti-mation

Résumé - Estimation du maximum de vraisemblance restreinte pour des modèles in-dividuels par dérivation de la vraisemblance. Cet article décrit une méthode d’estimation

du maximum de vraisemblance restreinte utilisant les dérivées

_première

et seconde de la vraisemblance. La méthode est basée sur une

_procédure

de

_{différenciation}

automatique ne

nécessitant pas l’inversion de

grandes

matrices. Elle constitue en

_fait

une extension de la

décomposition

de

_{Cholesky appliquée}

à une matrice. On utilise un

_paramétrage

_qui

trans-forme

le

_{problème d’optimisation}

avec _{contrainte que} soulève l’estimation des _composantes

de variance en un

_problème

sans _contrainte, ce _qui rend

_possible

l’utilisation

_{d’algorithmes}

de

_{Newton-Raphson}

ou

_apparentés.

Les calculs sont illustrés sur un

_{exemple numérique.}

Plusieurs

_algorithmes,

de type

Newton-Raphson

ou selon la méthode des _{scores, sont}

ap-pliqués

à

_l’analyse

de données sur bovins à viande. Ces

_algorithmes

sont

_comparés

entre

eu! _{et par} ailleurs

comparés

à un

algorithme

sans dérivation.

maximum de vraisemblance restreinte

_/

dérivée

_/

_algorithme

_/

estimation de

composante de variance *

INTRODUCTION

Maximum likelihood estimation of

_(co)variance

_components

_generally

_requires

the numerical solution of a constrained nonlinear

_optimisation

problem

(Harville,

1977).

Procedures to locate the minimum or maximum of a function are classified

_according

to the amount of information from derivatives of the function

utilised;

see, for

instance,

Gill et al

_(1981).

Methods

_using

both first and second derivatives are

fastest to _{converge, often}

_{showing quadratic}

_convergence, while search

_algorithms

not

_relying

on derivatives are

generally

slow, ie, require

_many iterations and function evaluations.

Early applications

of restricted maximum likelihood

_(REML)

estimation to

animal

_breeding

data used a Fisher’s method of

scoring

type

algorithm, following

the

_original

paper

by

Patterson and

_Thompson

₍₁₉₇₁₎

and

_Thompson

_(1973).

This

_requires

_expected

values of the second derivatives of the likelihood to be

evaluated,

which

proved computationally

highly demanding

for all but the

_simplest

analyses.

Hence

_{expectation-maximization}

_(EM)

_type

_{algorithms gained popularity}

and found

_widespread

use for

_{analyses fitting}

a sire model.

_Effectively,

these use first derivatives of the likelihood function.

_Except

for

_special

cases,

however,

they

required

the inverse of a matrix of size

_equal

to the number of random effects

_fitted,

eg, number of sires times number of

traits,

which

_severely

limited the size of

_analyses

feasible.

For

_analyses

under the animal

_model,

Graser et al

₍₁₉₈₇₎

thus

_proposed

a

derivative-free

_algorithm.

This

_only

requires

factorising

the coefficient matrix of the mixed-model

_equations

rather than

_{inverting it,}

and can be

_{implemented efficiently}

using

sparse matrix

techniques. Moreover,

it is

readily

extendable to animal models

including

additional random effects and multivariate

_analyses

_(Meyer,

_1989,

_1991).

Multi-trait animal model

analyses

fitting

additional random effects

_using

a

derivative-free

_algorithm

have been shown to be feasible.

_However,

_they

are

com-putationally highly demanding,

the number of likelihood evaluations

_required

in-creasing

exponentially

with the number of

_(co)variance

_components

to be estimated

simultaneously.

Groeneveld et al

_(1991),

for

_instance,

_reported

that 56 000 evalu-ations were

_required

to reach a

change

in likelihood smaller than

10-

7

when

esti-mating

60 covariance

_components

for five traits. While

_judicious

choice of

_starting

values and search

_strategies

_(eg,

_temporary

maximisation with

_respect

to a subset

of the

parameters

only)

together

with

_exploitation

of

_special

features of the data

structure

_might

reduce demands

_markedly

for individual

_analyses,

it remains true that derivative-free maximisation in

_high

dimensions _{is very} slow to _converge.

This makes a case for REML

_algorithms

_using

derivatives of the likelihood

for

_{multivariate,}

multidimensional animal model

_analyses.

Misztal

₍₁₉₉₄₎

_recently

presented

a

_comparison

of rates of _convergence of derivative-free and derivative

algorithms, concluding

that the latter had the

_potential

to be faster in almost all _{cases, in}

_particular

that their _{convergence rate}

_depended

little on the number of traits considered.

_Large-scale

animal model

_{applications using}

an EM

_type

algorithm

(Misztal, 1990)

or even a method of

_scoring

_algorithm

_{(Ducrocq, 1993)}

have been

_{reported, obtaining}

the

_large

matrix inverse

_(or

its

_trace)

_{required by}

REML estimation under an animal model

_using

first and second derivatives of the

likelihood

_{function, computed}

without

inverting large

matrices.

DERIVATIVES OF THE LIKELIHOOD

Consider the linear mixed model

where y,

b,

u and e denote the vectors of

_{observations,}

fixed

_effects,

random effects and residual errors,

respectively,

and X and Z are the incidence matrices

pertaining

to b and u. Let

V(u)

=

G,

V(e)

= R and

Cov(u,e’)

=

_0,

_so that

_V(y)

= V =

ZGZ’

₊ R.

Assuming

a multivariate normal

distribution, ie,

y -

N(Xb, V),

the

log

of the REML likelihood

(G)

is

(eg,

Harville,

1977)

where X* denotes a full-rank submatrix of X.

REML

_algorithms

_using

derivatives have

_generally

been derived

_{by differentiating}

!2!.

However,

as outlined

_previously

(Graser

et

_{al, 1987; Meyer,}

_1989),

_{log L}

can be

rewritten as

where C is the coefficient matrix in the mixed-model

_equations

_(MME)

_pertaining

to

_[1]

_(or

a full rank submatrix

thereof),

and P is a

matrix,

Alternative forms of the derivatives of the likelihood can then be obtained

by differentiating

[3]

instead of

_!2!.

Let 0 denote the vector of

_parameters

to be estimated with elements

9z ,

i =

_{1, ... ,}

_p. The first and second

partial

derivatives of the

_log

likelihood are then

Graser et al

₍₁₉₈₇₎

show how the last two terms in

_!3!,

_log

_ICI

and

_y’Py,

can

be evaluated in a

_general

_way for all models of form

[1]

by carrying

out a series

of Gaussian elimination

_steps

on the coefficient matrix in the MME

_augmented

_by

the vector of

_right-hand

sides and a

quadratic

in the data vector.

_Depending

on the

model of

_analysis

and structure of G and

_R,

the other two terms

_required

in

_!3!,

1991),

generally

requiring only

matrix

_{operations proportional}

to the number of traits considered. Derivatives of these four terms can be evaluated

_analogously.

Calculating

logiC!

and

_y’Py

and their derivatives

The mixed-model matrix

_(MMM)

or

_augmented

coefficient matrix

_pertaining

to

[1]

is

where r is the vector of

_right-hand

sides in the MME.

Using general

matrix

_results,

the derivatives of

_log

_{C !}

are

Partitioned matrix results

_give

_log

_IMI

=

log

!C!

₊

log(y’Py),

ie,

(Smith, 1995)

This

_gives

derivatives

Obviously,

these

_expressions

_{([7], [8], [10]}

and

_[11])

_involving

the inverse of the

_large

matrices M and C are

_{computationally}

intractable for _any sizable animal model

_analysis.

However,

the Gaussian elimination

procedure

with

diagonal

pivoting

advocated

_by

Graser et al

₍₁₉₈₇₎

is

_only

one of several _ways to ’factor’

a matrix. An alternative is a

_Cholesky

_{decomposition.}

This lends itself

_readily

to the solution of

_large

_positive

definite

_systems

of linear

_equations

_using

sparse

matrix

_storage

schemes.

_Appropriate

Fortran routines are

_given,

for

_instance,

_by

George

and Liu

₍₁₉₈₁₎

and have been used

_successfully

in derivative-free REML

for j

>

_i)

denote the

_Cholesky

factor of

_{M, ie,}

M = LL’. The determinant of _a

triangular

matrix is

_simply

the

_product

of its

_diagonal

elements.

_Hence,

with M

denoting

the size of

_M,

and

_from

_I

Smith

₍₁₉₉₅₎

describes

_algorithms,

outlined

below,

which allow the derivatives of the

_Cholesky

factor of a matrix to be evaluated while

_carrying

out the

_{factorisation,}

provided

the derivatives of the

_original

matrix are

specified.

Differentiating

[13]

and

_[14]

then

_gives

the derivatives of

_log

_ICI

and

_y’Py

as

simple

functions of the

_diagonal

elements of the

_Cholesky

matrix and its derivatives.

Calculating

logIRI

and its derivatives

Consider a multivariate

_analysis

for _q traits and let y be ordered

according

to traits within animals.

_Assuming

that error covariances between measurements on

different animals are _zero, R is

_{blockdiagonal}

for

_animals,

where is N the number of animals which have

_records,

_{and E}

₊

denotes the direct matrix sum

_(Searle,

_1982).

Hence

_log

_IRI

as well as its derivatives can be determined

by

considering

one animal at a time.

combinations of traits recorded

_(assuming

_single

records _per

_trait),

eg, W = 3 for

q = 2 with combinations trait 1

only,

trait 2

only

and both traits. For animal i which has combination of traits w,

R

i

is

equal

to

_E

_w

_,

the submatrix of E obtained

by deleting

rows and columns

_pertaining

to

_missing

records. As outlined

_{by Meyer}

(1991),

this

_gives

where

_N

_w

_represents

the number of animals

_having

records for combination of traits

w.

_{Correspondingly,}

Consider the case where the

parameters

to be estimated are the

_(co)variance

components

due to random effects and residual errors

(rather

_than,

for

_example,

p

heritabilities and

_{correlations),}

so that V is linear in

0,

ie,

V =

£

9

j0V/

0

9z.

i=l

Defining

with elements

d

kL

=

_1,

if the klth element

ofE

w

is

equal

_to

9z ,

and

dk!

= 0

otherwise,

this then

_gives

Let

_e!

denote the rsth element of

_Ew

_l

_.

_{For ()}

_i

= _ekl and

9

j

=

e&dquo;, n

,

[23]

and

[24]

then

_simplify

to

For

q = 1 and R =

(

T

2j,

[25]

and

[26]

become

N

QE

2

and

-N(j

E4

,

respectively

(for

o

i

= oj

=

U2

E ).

_{Extensions for models with}

_repeated

records are

straightforward.

Hence,

once the inverses of the matrices of residual covariances for all combination

of numbers of traits recorded

_occurring

in the data have been obtained

_(of

maximum

size

_equal

to the maximum number of traits recorded _per

_animal,

and also

required

to set _up the

_MMM),

evaluation of

_log

_!R!

and its derivatives

requires

only

scalar

manipulations

in addition.

Calculating

loglGI

and its derivatives

Terms

_arising

from the covariance matrix of random

_{effects, G,}

can often be

determined in a similar _way,

exploiting

the structure of G. This

depends

on the

random effects fitted.

Meyer

(1989, 1991)

describes

_log

_IGI

for various cases.

Define T with elements

_t

_ij

of size rq x _rq as the matrix of covariances between

random effects where r is the number of random factors in the model

(excluding

e).

For

_{illustration,}

let u consist of a vector of animal

_genetic

effects a and some uncorrelated additional random

_effect(s)

c with

_N

_c

levels _per

trait, ie,

u’ =

(a’c’).

In the

_simplest

_case, a consists of the direct additive

genetic

effects for each animal

and

trait, ie,

it has

_length

_qN

_A

where

_N

_A

denotes the total number of animals in the

_{analysis, including}

parents

without records. In other cases, a

might

include a

second

_genetic

effect for each animal and

_trait,

such as a maternal additive

_genetic

effect,

which _may be correlated to the direct

_genetic

effects. An

_example

for c is a common environmental effect such as a litter effect.

With a and c

_{uncorrelated,}

T can be

partitioned

into

corresponding diagonal

blocks

T

A

and

_T

_C

_,

so that

where A is the numerator

relationship

between

_{animals, F,}

often assumed to be the

_identity

_matrix,

describes the correlation structure

_amongst

the levels of c, and

x denotes the direct matrix

_product

_{(Searle, 1982).}

This

_gives

_(Meyer,

₁₉₉₁₎

Noting

that all

₈

₂

_T/8()i8()j

= 0

(for

V linear in

0),

derivatives are

where

DA

=

8T

A/

a9

i

and

D!

=

8T

c/

8()

i

are

_again

matrices with elements 1

if tkl =

()

i

and zero otherwise. As

above,

all second derivatives

for O

i

and

9

j

not

pertaining

to the same random factor

(eg, c)

or two correlated factors

_(such

as

direct and maternal

genetic

effects)

are zero.

Furthermore,

all derivatives of

log

G ) I

with

_respect

to residual covariance

components

are zero.

Further

_{simplifications analogous}

to

_[25]

and

_[26]

can be derived. For

instance,

random effects

_(r

=

1),

T is the matrix of additive

_genetic

covariances

ai! with

i, j = 1, ... , q. For O

i

= _ax!

and O

j

=

a mn

, this

gives

with ars

_denoting

the rsth element of

T-

1

.

_For

_{q = 1} and _all =

QA

,

[31]

and

[32]

reduce to

_N

_A

_(jA

₂

and

_-N

_A

_(jA

₄

_,

_{respectively.}

Derivatives of the mixed model matrix

As

_emphasised

_above,

calculation of the derivatives of the

_Cholesky

factor of M

_requires

the

_{corresponding}

derivatives of M to be evaluated.

_Fortunately,

these have the same structure as M and can be evaluated while

_setting

_up

_M,

_replacing

G and R

_by

their derivatives.

For O

i

_{and O}

_j

equal

to residual

_{(co)variances,}

the derivatives of M are of the

form

with

Q

R

standing

in turn for

and

for first and second

_{derivatives, respectively.}

As outlined

_above,

R is

blockdiagonal

for animals with submatrices

E

w

.

Hence,

matrices

_Q

_R

have the same structure with submatrices

and

_(for

V linear in 0 so that

é

P

R/8()/}()j =

0)

Consequently,

the derivatives of M with

_respect

to the residual

_{(co)variances}

can

be _{set up} in the same _way as the ’data

part’

of M. In addition to

_calculating

the

matrices

_Ew

_l

for the W combination of records per animal

occurring

in the

data,

all derivatives of the

_E-

₁

for residual

_components

need to evaluated. The extra calculations

_{required, however,}

are

_{trivial, requiring}

matrix

_{operations proportional}

to the maximum number of records _per animal

_only

to obtain the terms in

_[36]

Analogously,

for O

i

and O

j

equal

to elements of

_T,

derivatives of M are

with

_Q

_G

standing

for

for first

_derivatives,

and

for second derivatives.

As

_above,

further

_{simplifications}

are

possible depending

on the structure of G. For

_instance,

for G as in

_[27]

and

_[j

₂

_G/å()

_J

_}()j

=

0,

and

_Qo

₂

₌

Expected

values of second derivatives of

_logc

Differentiating

[2]

gives

second derivatives of

_logc

with

_expected

values

_{(Harville, 1977)}

Again,

for V linear in

_0,

_(9’VlaOiaO

_j

= 0. From

[5]

and

noting

that

aPla0

i

=

Hence, expected

values of the second derivatives are

_essentially

_(sign

ignored)

equal

to the observed values minus the contribution from the

data,

and thus can be

evaluated

_analogously.

With second derivatives of

_y’Py

not

_{required, computational}

requirements

are reduced somewhat as

_only

the first M — 1 rows of

82M/8()i8()j

need to be evaluated and factored.

AUTOMATIC DIFFERENTIATION

Calculation of the derivatives of the likelihood as described above relies on the

fact that the derivatives of the

_Cholesky

factor of a matrix can be obtained

’automatically’, provided

the derivatives of the

_original

matrix can be

specified.

Smith

₍₁₉₉₅₎

describes a so-called forward

_{differentiation,}

which is a

straight-forward

_expansion

of the recursions

_employed

in the

_Cholesky

factorisation of a

matrix M.

_Operations

to determine the latter are

_typically

carried out

sequentially

by

rows. Let

_L,

of size

_N,

be initialised to M.

_First,

the

_pivot

_(diagonal

element which must be

_greater

than an

_operational

_zero)

is selected for the current row k.

Secondly,

the

off-diagonal

elements for the row

(’lead column’)

are

adjusted

( L

_jk

for j

= k ₊

1, ... ,

N),

and

thirdly

the elements in the

remaining

_part

of L

(L2!

for

j

=

k+1, ... ,

N and i =

_{j, ... ,}

N)

are modified

_(’row

operations’).

After all N rows have been

_processed,

L contains the

_Cholesky

factor of M.

Pseudo-code

_given

_by

Smith

₍₁₉₉₅₎

for the calculation of the

_Cholesky

factor and its first and second derivatives is summarised in table I. It can be seen that the

operations

to evaluate a second derivative

_require

the

_respective

elements of

the two

_{corresponding}

first derivatives. This

_imposes

severe constraints on the

memory

requirements

of the

_algorithm.

While it is most efficient to evaluate the

Cholesky

factor and all its derivatives

_together,

considerable _space can be saved

_by

computing

the second derivatives one at a time. This can be done

_{by holding}

all

the first derivatives in memory, or, if core _space is the

_limiting

factor, storing

first

derivatives on disk

_(after

_evaluating

them

_individually

as

_well)

and

_reading

in

_only

the two

_{required. Hence,}

the minimum memory

requirement

for REML

_using

first and second derivatives is 4 x

_{L, compared}

to L for a derivative-free

_algorithm.

Smith

₍₁₉₉₅₎

stated

_{that, using}

forward

_{differentiation,}

each first derivative

required

not more than twice the work

required

to evaluate

_log

G

_only,

and that the work needed to determine a second derivative would be at most four times that

to calculate

log

G.

In

_addition,

Smith

₍₁₉₉₅₎

described a ’backward differentiation’

_scheme,

so

named because it reverses the order of

_steps

in the forward differentiation. It is

applicable

for cases where we want to evaluate a scalar function of

_L,

f (L),

in our case

_{log I C + y’Py}

which is a function of the

diagonal

elements of L

(see

[13]

and

!14!).

It

_{requires computing}

a

(lower

triangular)

matrix W

_which,

on

_completion

of

the backward

differentiation,

contains the derivatives of

_{f (L)}

with

respect

to the elements of M. First derivatives of

_{f (L)}

can then be evaluated one at a time as

tr(W 8M/ 8()r)’

The

pseudo-code given

by

Smith

₍₁₉₉₅₎

for the backward differentiation is shown in table II. Calculation of W

_requires

about twice as much work as one likelihood

evaluation, and,

once W is

_evaluated,

_calculating

individual derivatives

_(step

3 in

differentiation

_requires

_only

somewhat more work than calculation of one derivative

by

forward differentiation. Smith

₍₁₉₉₅₎

also described the calculation of second derivatives

_by

backward differentiation

_(pseudo-code

not shown

_here).

_Amongst

other

_{calculations,}

this involves one evaluation of a matrix W as described

_above,

for each

parameter

and

_requires

another work array of size L in addition to _space to store at least one matrix of derivatives of M. Hence the minimum memory

requirement

for this

_algorithm

is 3 x _{L + M}

_(M

and L

_{differing by}

the fill-in created

_during

the

_{factorisation).}

Smith

₍₁₉₉₅₎

claimed that the total work

_required

to evaluate all second derivatives for p

parameters

was no more than

6p

times that for a likelihood evaluation.

MAXIMISING THE LIKELIHOOD

Methods to locate the maximum of the likelihood function in the context of

variance

_component

estimation are

_reviewed,

for

_instance,

_by

Harville

(1977)

and

Searle et al

_(1992;

Chapter

8).

Most utilise the

_gradient

vector,

ie,

vector of first derivatives of the likelihood

function,

to determine the direction of search.

Using

second derivatives

One of the oldest and most

_widely

used methods to

_optimise

a non-linear function

is the

_{Newton-Raphson}

_(NR)

_algorithm.

It

_requires

the Hessian matrix of the

function,

ie,

the matrix of second

partial

derivatives of the

_(log)

likelihood with

where

H’

=

{å2log£/å()iå()j}

and g’

=

10logLI,90

il

are the Hessian matrix

and

_gradient

vector of

_log

_£

_,

_{respectively,}

both evaluated at 0 =

0!.

While the NR

_algorithm

can be

_quick

to _{converge, in}

_particular

for functions

resembling

a

quadratic function,

it is known to be sensitive to poor

starting

values

_{(Powell, 1970).}

Unlike other

_algorithms,

it is not

_guaranteed

_{to converge}

_{though global}

_convergence has been shown for some cases

_using

iterative

_partial

maximisation

_(Jensen

et

_al,

1991).

In

_practice,

so-called extended or modified NR

algorithms

have been found to

be more successful. Jennrich and

Sampson

(1976)

suggested

step

halving,

applied

successively

until the likelihood is found to

_increase,

to avoid

_{’overshooting’.}

More

generally,

the

change

in estimates for the tth iterate in

_[46]

is

_given

_by

for the extended

NR,

B’

=

—T!(H!) !,

where

T

’

is a

step-size

scaling

factor. The

optimum

for

T

t

can

be determined

_readily

as the value which results in the

largest

increase in

_{likelihood, using}

a one-dimensional maximisation

technique

(Powell,

1970).

This relies on the direction of search

_{given by}

H-

l

g

generally

being

a ’good’

direction and

that,

for -H

positive definite,

there is

_always

a

_step-size

which will

increase the likelihood.

Alternatively,

the use of

has been

_suggested

_{(Marquardt, 1963)}

to

_improve

the

_performance

of the NR

algorithm.

This results in a

_step

intermediate between a NR

_step

(!

=

0)

and

a method of

_steepest

ascent

_{step (}

_K

_large).

_{Again, !}

can be chosen to maximise the

increase in

_{log G, though}

for

_large

values of K the

step

size is

_small,

so that there is no need to include a search

_step

in the iteration

(Powell, 1970).

Often

_expected

values of the second derivatives of

_log

_G

are easier to calculate than the observed values.

_{Replacing -H}

by

the information matrix

results in Fisher’s method of

scoring

(MSC).

It can be extended or modified in the

same _way as the NR

algorithm

(Harville, 1977).

Jennrich and

Sampson

(1976)

and

Jennrich and Schluchter

₍₁₉₈₆₎

compared

NR and

_{MSC, showing}

that the MSC was

generally

more robust

against

a _poor choice of

_starting

values than the

NR,

though

it tended to

_require

more iterations.

They

thus recommended a scheme

_using

the

MSC

initially

and

_switching

to NR after a few rounds of iteration when the increase

in

_log

G between

steps

was less than one.

Using

first derivatives

only

Other

methods,

so-called variable-metric or

Quasi-Newton procedures, essentially

use the same

_strategies,

but

replace

B

by

an

_{approximation}

of the Hessian matrix.

An

_interesting

variation has

_recently

been

_presented

_by

Johnson and

Thompson

(1995).

Noting

that the observed and

expected

information were of

_{opposite sign}

and differed

_{only by}

a term

_involving

the second derivatives of

_{y’Py (see}

_[5]

and

[45]),

they

suggested

using

the _average of observed and

_expected

information

_(AI)

to

_approximate

the Hessian matrix. Since it

requires only

the second derivatives of

y’Py

to be

evaluated,

each iterate is

_{computationally considerably}

less

_demanding

than a ’full’ NR or MSC

_step,

and the same modifications as described above for the

NR

_(see

_[47]

and

_[48])

can be

_applied.

Initial

_comparisons

of the rate _{of convergence} and

_computer

time

_required

showed the AI

_algorithm

to be

_highly

_advantageous

over both derivative-free and

EM-type

procedures

(Johnson

and

Thompson,

1995).

Constraining

parameters

All the

_Newton-type

_algorithms

described above

perform

an unconstrained

optimisation.

To

_estimating

_(co)variance

_components,

_however,

we

_require

variances

be

_{non-negative,}

correlations to be in the range from -1 to 1

and,

for more than

two

_traits,

for them to be consistent with each

_other;

more

generally,

estimated

covariance matrices need to be

_(semi-)

_positive

definite. As shown

_by

Hill and

Thompson

(1978),

the

_probability

of

obtaining

parameter

estimates out of bounds

depends

on the

magnitude

of the correlations

(population values)

and increases

rapidly

with the number of traits

_considered,

in

_particular

for

_genetic

covariance

matrices.

For univariate

_analyses,

a common

approach

has been to set

_negative

estimates of variance

_components

to zero and continue

_iterating.

Partial

_sweeping

of B to handle

_{boundary constraints, monitoring}

and

_restraining

the size of

_pivots

relative to the

_{corresponding}

_(original)

diagonal

elements has been recommended

_(Jennrich

and

Sampson, 1976;

Jennrich and

_Schluchter,

_1986).

Harville

_(1977;

section

_6.3)

dis-tinguished

between three

_types

of

_techniques

to

_modify

unconstrained

_optimisation

procedures

to accommodate constraints on the

_parameter

_space.

Firstly, penalty

techniques

operate

on a function which is close to

log L

_except

at the boundaries

where it assumes

_large

_negative

values which

effectively

serve as a

_barrier,

_deflecting

a further search in that direction.

Secondly, gradient

projection techniques

are

suit-able for linear

inequality

constraints.

_Thirdly,

it _may be feasible to transform the

parameters

to be estimated so that maximisation on the new scale is unconstrained.

Box

₍₁₉₆₅₎

demonstrated for several

examples

that the

_{computational}

effort to

solve constrained

problems

can be reduced

_{markedly by}

_eliminating

constraints so

that one of the more

powerful

unconstrained

methods,

preferably

with

quadratic

convergence, can be

_employed.

For univariate

_analysis,

an obvious _way to eliminate

non-negativity

constraints is to maximise

_{log L}

with

_respect

to standard deviations and to _square these on _convergence

(Harville,

1977).

Alternatively,

we could

estimate

_logarithmic

values of variance

_components

instead of the variances. This

seems

_preferable

to

_taking

_square roots

_where,

on

_{backtransforming,}

a

_largish

negative

estimate

_might

become a substantial

_positive

estimate of the

corresponding

variance. Harville

₍₁₉₇₇₎

cautioned however that such transformations may result in the introduction of additional

_{stationary points}

on the likelihood

surface,

and

thus should be used

_only

in

_conjunction

with

_{optimisation techniques ensuring}

an

Further motivation for the use of transformations has been

_provided

_by

the

scope to reduce

_{computational}

effort or to

_improve

_convergence

_by

_making

the

shape

of the likelihood function on the new scale more

_quadratic.

For multivariate

analyses

with one random effect and

_equal

_design

_matrices,

for

_instance,

a canonical

transformation allows estimation to be broken down into a series of

_{corresponding}

univariate

_analyses

_{(Meyer, 1985);}

see Jensen and Mao

₍₁₉₈₈₎

for a review. Harville and Callanan

₍₁₉₉₀₎

considered different forms of the likelihood for both

NR and MSC and demonstrated how

_they

affected _convergence behaviour. In

particular,

a ’linearisation’ was found to reduce the number of iterates

_required

to reach _convergence

_{considerably. Thompson}

and

_Meyer

₍₁₉₈₆₎

showed how a

reparameterisation

aimed at

_making

variables less correlated could

_speed

up an

expectation-maximisation

algorithm dramatically.

For univariate

_analyses

of

_repeated

measures data with several random

_effects,

Lindstrom and Bates

₍₁₉₈₈₎

_suggested

maximisation of

_{log £}

with

_respect

to the non-zero elements of the

_{Cholesky decomposition}

of the covariance matrix of random effects in order to remove constraints on the

parameter

space and

to

_{improve stability}

of the NR

algorithm.

In

_{addition, they}

chose to estimate

the error variance

directly,

_operating

on the

_profile

likelihood of the

_remaining

parameters.

For several

_examples,

the authors found consistent _convergence of the

NR

_algorithm

when

implemented

this _way, even for an

_{overparameterised}

model.

Recently,

Groeneveld

₍₁₉₉₄₎

examined the effect of this

_{reparameterisation}

for

large-scale

multivariate

_analyses

_using

derivative-free

_{REML, reporting}

substantial

improvements

in

_speed

of _convergence for both direct search

_{(downhill Simplex)}

and

_Quasi-Newton

_algorithms.

IMPLEMENTATION

Reparameterisation

For our

analysis,

_parameters

to be estimated are the non-zero elements of the lower

triangle

of the two covariance matrices E and

_T,

with T

_potentially

_consisting

of

independent diagonal

blocks,

depending

on the random effects fitted. Let

The

_{Cholesky decomposition}

of U has the same structure

Estimating

the non-zero elements of matrices

_L

_u.

then ensures

_positive

definite matrices

_U

_r

on

_transforming

back to the

_original

scale

_(Lindstrom

and

Bates,

1988).

However,

for the

_U

_r

_representing

covariance

_matrices,

the ith

_diagonal

element of

L

is

_suggested

to

_apply

a

_{secondary transformation, estimating}

the

logarithmic

values

of these

_diagonal

elements and

_thus,

as discussed above for univariate

_analyses

of

variance

_components,

_{effectively forcing}

them to be

_greater

than zero.

Let v denote the vector of

parameters

on the new scale. In order to maximise

_{log G}

with

_respect

to the elements of _v, we need to transform its derivatives

_accordingly.

Lindstrom and Bates

₍₁₉₈₈₎

describe

_briefly

how to obtain the

_gradient

vector and Hessian matrix for a

_Cholesky

matrix

_L

_Ur

from those for the matrix

U

r

(see

also

corrections,

Lindstrom and Bates

_(1994)).

For the ith element of v,

More

_generally,

for a one-to-one transformation

_{(Zacks, 1971)}

where J with elements

_8()d8v

_j

is the Jacobian of 9 with respect to v.

Similarly,

with the first

part

of

_[53]

_equal

to the ith element of

_{(0J’/0vj )(0 10g £/09k)}

_and

the second

_part

_equal

to

_ijth

element of

_{J’ {}

_ô

_{2log L/ Ô()}

_i

_Ô()

_j

_}J.

Consider one covariance matrix

_U

_r

at a

_time,

dropping

the

subscript

r for

convenience,

and let u,t and

l

wx

denote the elements of U and

L

U

,

respectively.

From U =

LL’,

it follows that

where

_{min(s, t)}

is the smaller value of s and t.

_Hence,

the

_ijth

element of J for

()i

= _Ust

and v

j

=

l

wx

is

For

_{w # x}

and

s,

t ! x, this is non-zero

_only

if at least one of s and t is

_equal

to

cases to consider :

For

_example,

for _q = 3 traits and six elements in B

(ull,

_{ulz, U13, uzz, U23} and

U33

)

and v

_(log(lil),

l

zi

,

l

si

,

),

log(<

22

1

32

,

log(l

33

))

Similarly,

the

_ijth

element of

_}

_j

_iåv

_V

_å

_k/

₍₎

₂

_{å

_(see

_[53])

for

9

k

=

u, t

, vi =

l

wx

and

1/! =

ly

z

is

Allowing

for the additional

_adjustments

due to the

_log

transformation of

diago-nals,

[56]

is non-zero in five cases

(for

_{w 7!}

t

_!

x and

_{x, t <}

w):

For the above

_example,

this

_gives

the first two derivatives of J

In some _cases, a covariance

_component

cannot be

_estimated,

for

_instance,

the error covariance between two traits measured on different sets of animals. On the

new

_parameter

_may be non-zero. This can be accommodated

by, again,

not

estimating

this

parameter

but

_calculating

its new value from the estimates of the

other

_parameters,

similar to the way a correlation is fixed to a certain value

_by

only

estimating

the

respective

variances and then

_calculating

the new covariance

’estimate’ from them.

For

_example,

consider three traits with the covariance between traits 2 and

3,

U23, which are not estimable.

Setting

u23 = 0

gives

a non-zero value on

the

_{reparameterised}

scale of

!32

=

-1

21

1

31

/1

22

.

Only

the other five

parameters

(lii, 1

21

, 131, 122,

and

_l

₃₃

₎

are then estimated

(ignoring

the

log

transformation of

diagonals

here for

_simplicity)

while an ’estimate’ of

l

32

is calculated from those of

1

21

,

l

31

and

_l

₃₃

_according

to the

_relationship

above. On

transforming

back to the

original

scale,

this ensures that the covariance between traits 2 and 3 remains fixed

at zero.

Sparse

matrix

storage

Calculation of

_{log £}

for use in derivative-free REML estimation under an animal model has been made feasible for

practically

sized data sets

_through

the use of

sparse matrix

storage

techniques.

These

_{adapt readily}

to include the calculation of derivatives of L. One

_possibility

is the use of so-called linked lists

(see,

for

_instance,

Tier and Smith

_(1989)).

_George

and Liu

₍₁₉₈₁₎

describe several other

_strategies,

using a ’compressed’

storage

scheme when

applying

a

_{Cholesky decomposition}

to a

large symmetric, positive definite,

sparse matrix.

Elements of the matrices of derivatives of L are subsets of elements of

L, ie,

exist

only

for non-zero elements of L. Thus the same

_system

of

_pointers

can be used for L

and all its

_{derivatives, reducing}

overheads for

storage

and calculation of addresses of individual elements

_(though

at the _expense of

_reserving

space for zero derivatives

corresponding

to non-zero

_l2!).

_Moreover,

schemes aimed at

_reducing

the ’fill-in’

during

the factorisation of

_M,

should also reduce

_{computational}

requirements

for

determining

derivatives.

Matrix

_storage

_required

in

_evaluating

derivatives of

_log

_£

can be considerable: for

p

parameters

to be

_estimated,

there are _p first and

p(p+1)/2

second

derivatives, ie,

up to

1!-p(p+3)/2

times as much _space as for

_{calculating log £ only}

can be

required.

Even for

analyses

with one random factor

(animals)

only,

this becomes

prohibitive

very

quickly, amounting

to a factor of 28

_for

q = 2 traits and p = 6

_parameters,

and 91 for q = 3 and p = 12.

However,

while matrices

aLla0

i

are needed to evaluate

second

derivatives,

matrices

₈

₂

_L/

₈

₍₎

_i

₈₍₎

_j

are not

_required

after

_a

₂

_{log ICI/}

₈

₍₎

_i8

₍₎

_j

and

₈

₂

_y’Py

_/

₈₍₎

_i

₈₍₎

_j

have been calculated from their

diagonal

elements.

_Hence,

while it is most efficient to evaluate all derivatives of each

_li!

_{simultaneously,}

second derivatives can be determined one at a time after L and its first derivatives have

been determined. This can be done

_by

_setting

_up and

_processing

each

8

2

M/8()

i

8()j

individually

thus

_reducing

_memory

_{required dramatically.}

Software

Cholesky

decomposition

of the covariance matrices

_(and

logarithmic

values of their

diagonals),

as described

_above,

to remove constraints on the

_parameter

_space. Prior to

_estimation,

the

ordering

of rows and columns 1 to M - 1 in M was

determined

_using

the minimum

_{degree re-ordering performed by George}

and Liu’s

(1981)

subroutine

_GENQMD,

and their subroutine SBMFCT was used to establish the

_symbolic

factorisation of M

_(all

M rows and

columns)

and the associated

compressed

storage

pointers, allocating

space for all non-zero elements of L. In

addition,

the use of the _average information matrix was

implemented.

However,

this was done

merely

for the

_comparison

of _{convergence rates} without

_making

use

of the fact that

_only

the derivatives of

y’Py

were

_required.

For each

_iterate,

the

optimal

step

size or

_scaling

factor

(see

[47]

and

!48!)

was

determined

_{by carrying}

out a

one-dimensional,

derivative-free search. This was done

using

a

_{quadratic approximation}

of the likelihood

surface, allowing

for _{up to} five

approximation

steps

per iterate. Other

techniques,

such as a

_{simple step-halving}

procedure,

would be suitable alternatives.

Both

_procedures

described

_by

Smith

₍₁₉₉₅₎

to _{carry out} the automatic

differenti-ation of the

_Cholesky

factor of a matrix were

implemented.

Forward differentiation

was set _up to evaluate L and all its first and second derivatives to be as efficient as

possible,

ie,

holding

all matrices of derivatives in core and

evaluating

them

simulta-neously.

In

_addition,

it was _{set up} to reduce _memory

_{requirements, ie,}

_calculating

the

_Cholesky

factor and its first derivatives

_together

and

_subsequently

_evaluating

second derivatives

_{individually.}

Backward differentiation was

implemented

storing

all first derivatives of M in core but

_setting

_up and

processing

one second derivative

of M at a time.

EXAMPLES

To illustrate the

_{calculations,}

consider the test data for a bivariate

analysis

given

by

Meyer

(1991).

Fitting

a

_simple

animal

model,

there are six

_parameters.

Table III summarises intermediate results for the likelihood and its

derivatives

for the

_starting

values used

by Meyer

(1991),

and

_gives

estimates from the first round of iteration for

_simple

or modified NR or MSC

algorithms.

Without

reparameterisation,

the

(unmodified)

NR

_{algorithm produced}

estimates out of bounds of the

_parameter

space, while the MSC

performed

quite

well for this first iterate.

Continuing

to use

expected

values of second derivatives

though,

estimates failed to _converge.

Figure

1 shows the

_{corresponding}

change

in

_log

G over rounds of iteration. For

this small

_example,

with a

_starting

value for the

additive

_genetic

covariance

(

QA12

)

very different from the eventual

estimate,

a NR or MSC

_algorithm

_optimising

the

step

size

_(see

_equation

_!47!)

failed to locate a suitable

_step

size

(which

increased

log

G)

in the first iterate.

_Essentially,

all

_algorithms

had reached about the same

point

on the likelihood surface

_by

the fifth

_iterate,

with _very little

changes

in

log £

in

_subsequent

rounds.

_Convergence

was considered achieved when the norm of the

gradient

vector was less than

10-

4

.

This was a rather strict criterion:

changes

in

estimates and likelihood values between iterates were

_usually

_very small before

likelihood evaluations were

_required

to reach _convergence

_using

the observed information

compared

to

_eight

iterates and 34 likelihood evaluations for the average

information while the MSC

_{(expected information)}

failed in this case.

_Optimising

the

_step

size

_(see

_!47)),

_{7 + 40,}

10 + 62 and 10 + 57 iterates + likelihood evaluations

were

_{required using}

the

observed, expected

and average

information,

respectively.

As illustrated in

_Figure

_2,

use of the

_’average

information’

yielded

a _very similar

convergence

pattern

to that of the NR

algorithm.

In

_{comparison, using}

an EM

algorithm

for this

_example,

80 rounds of iteration were

required

before the

_change

in

_{log £}

between iterates was less than

10-

4

and 142 iterates were needed before

the average

changes

in estimates was less than

0.01%.

a

( T A ij

: additive

genetic

covariances; (T Eij : residual

covariances;

b

log

likelihood: values

given

differ from those

_{given by Meyer}

(1991)

_by

a constant offset of 3.466 due to

absorp-tion of

_single-link

_parents without records

_{(‘pruning’).}

_Components

of first derivatives of

_{log ,C;}

see text for definition.

d

First

derivative of

_log

G with _respect to

_B

_i

_.

e Second

derivative of

_{log £}

with _respect to

Bi.

f

Expected

value of second derivative of

_{log £}

with respect to

_B

_i

_.

second derivatives of

_log

G with _respect to

_O

_i

and

₀

_j

_:

observed values

Table IV

_gives

characteristics of the data structure and model of

_analysis

for

applied examples

of multivariate

analyses

of beef cattle data. The first is a bivariate

analysis

fitting

an animal model with maternal

_permanent

environmental effects as an additional random

_{effect, ie, estimating}

nine covariance

_components.

The second

shows the

_analysis

of three

_weight

traits in Zebu Cross cattle

_(analysed

_previously;

see

_Meyer

(1994a))

under three models of

analysis,

_estimating

_{up to} 24

_parameters.

For each data set and

model,

the

_{computational requirements}

to obtain

deriva-tives of the likelihood were determined

_using

forward and backward differentiation

!NR:

Newton-Raphson;

MSC: method of

scoring;

MIX:

_Starting

as

_{MSC, switching}

to NR when

_change

in

_log

likelihood between iterates

_drops

below

1.0;

and DF: derivative-free

algorithm.

b

CB:

Cannon bone

length;

HH:

_{hip height;}

WW:

weaning weight;

YW:

yearling weight;

FW: final

weight.

!1:

_simple

animal

_model;

2: animal model

_fitting

dams’ _permanent environmental

_effect;

and 5: animal model

_fitting

both

_genetic

and

permanent environmental effects.

d

A:

Forward

_{differentiation, processing}

all derivatives

simultaneously;

B: Forward

_{differentiation, processing}

second derivatives one at a _time;

C: Backward differentation. 0: No

_{reparameterisation,}

_unmodified;

S:

_{reparameterised}

scale, optimising

step

size,

see

[47]

in _text; T:

_{reparameterised scale, modifying diagonals}

as described

_by

Smith

_(1995).

If the _memory available

_allowed,

forward

differentia-tion was carried out for all derivatives

_{simultaneously}

and

_considering

one matrix

8

2

M/

8

()

i

8()j

at a time. Table IV

_{gives computing}

times in minutes for

calcula-tions carried out on a DEC

Alpha Chip

(DIGITAL)

machine

_running

under

OSF/1

I

(rated

at about 200

_Mflops).

Clearly,

the backward differentiation is most

compet-itive,

with the time

_required

to calculate 24 first and 300 second derivatives

_being

’only’

about 120 times that to obtain

_{log L only.}

Starting

values used were

_{usually ’good’,}

_using

estimates of variance

_components

from

_{corresponding}

univariate

_{analyses throughout}

and

_deriving

initial _guesses for covariance

components

from these and literature values for the correlations concerned. A maximum of 20 iterates was carried

_out,

_applying

a _very

_stringent

convergence criterion of a

change

in

log L

less than

10-

6

between iterates or a value

for the norm of the

_gradient

vector of less than

10-

4

as above.

The numbers of iterates and additional likelihood evaluations

_required

for each

algorithm

given

in table IV show small and inconsistent differences between them. On the

_{whole, starting}

with a MSC

_algorithm

and

_switching

to NR when the

_change

in

_log

G became small and

_{applying Marquardt’s}

₍₁₉₆₃₎

modification

_{(see !48!)}

tended to be more robust

_(ie,

achieve convergence when other

algorithm(s) failed)

for

_starting

values not too close to the eventual estimates.

_{Conversely, they}

tended

to be slower to _converge than an NR

_optimising

_step

size

(see

!47))

for

_starting

values very close to the estimates.

_However,

once the derivatives of

log

G have

been

_evaluated,

it is

_{computationally}

_{relatively inexpensive}

_(requiring

_only

_simple

likelihood

_evaluations)

to compare and switch between

_algorithms,

ie,

it should be feasible to select the best

_procedure

to be used for each

_{analysis individually.}

Comparisons

with EM

_algorithms

were not carried out for these

_examples.

However,

Misztal

₍₁₉₉₄₎

claimed that each _{sparse matrix} inversion

_required

for

an EM

_step

took about three times as

long

as one likelihood evaluation. This

would mean that each iterate

_using

second derivatives for the three-trait

_analysis

estimating

24

_highly

correlated

_{(co)variance components}

would

_require

about the

same time as 40 EM

_iterates,

or that the second derivative

_algorithm

would be

advantageous

if the EM

_algorithm

_required

more than 462 iterates to _converge.

DISCUSSION

For univariate

_analyses,

_Meyer

_(1994b)

found no

advantage

in

_{using Newton-type}

algorithms

over derivative-free REML.

_{However, comparing}

total _cpu time per

analysis, using

derivatives _appears to be

_{highly advantageous}

over a

derivative-free

_algorithm,

for multivariate

_analyses,

the more so the

larger

the number of

parameters

to be estimated.

_Furthermore,

for

_{analyses involving larger}

numbers of

parameters (18

or

_24),

the derivative-free

_algorithm

converged

several times to a

local

_maximum,

a

_problem

observed

_{previously by}

Groeneveld and Kovac

(1990),

while the

_Newton-type

_{algorithm appeared}

not be affected.

Modifications of the NR

algorithm,

combined with a local search for the

_optimum

step

size,

improved

its _convergence rate. This was achieved

_{primarily by}

safe-guarding against

steps

in the

_’wrong’

direction in

_{early iterates,}

thus

_making

the

search for the maximum of the likelihood function more robust

_against

bad

_starting